Integrand size = 28, antiderivative size = 276 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\frac {b^2 f m n^2 \log (x)}{2 e}-\frac {b f m n \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {f m \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {b^2 f m n^2 \log \left (e+f x^2\right )}{4 e}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {b^2 f m n^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{4 e}+\frac {b f m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{2 e}+\frac {b^2 f m n^2 \operatorname {PolyLog}\left (3,-\frac {e}{f x^2}\right )}{4 e} \]
1/2*b^2*f*m*n^2*ln(x)/e-1/2*b*f*m*n*ln(1+e/f/x^2)*(a+b*ln(c*x^n))/e-1/2*f* m*ln(1+e/f/x^2)*(a+b*ln(c*x^n))^2/e-1/4*b^2*f*m*n^2*ln(f*x^2+e)/e-1/4*b^2* n^2*ln(d*(f*x^2+e)^m)/x^2-1/2*b*n*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x^2-1/ 2*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x^2+1/4*b^2*f*m*n^2*polylog(2,-e/f/x ^2)/e+1/2*b*f*m*n*(a+b*ln(c*x^n))*polylog(2,-e/f/x^2)/e+1/4*b^2*f*m*n^2*po lylog(3,-e/f/x^2)/e
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 946, normalized size of antiderivative = 3.43 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=-\frac {-12 a^2 f m x^2 \log (x)-12 a b f m n x^2 \log (x)-6 b^2 f m n^2 x^2 \log (x)+12 a b f m n x^2 \log ^2(x)+6 b^2 f m n^2 x^2 \log ^2(x)-4 b^2 f m n^2 x^2 \log ^3(x)-24 a b f m x^2 \log (x) \log \left (c x^n\right )-12 b^2 f m n x^2 \log (x) \log \left (c x^n\right )+12 b^2 f m n x^2 \log ^2(x) \log \left (c x^n\right )-12 b^2 f m x^2 \log (x) \log ^2\left (c x^n\right )+12 a b f m n x^2 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 b^2 f m n^2 x^2 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-6 b^2 f m n^2 x^2 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+12 b^2 f m n x^2 \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+12 a b f m n x^2 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 b^2 f m n^2 x^2 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-6 b^2 f m n^2 x^2 \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+12 b^2 f m n x^2 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 a^2 f m x^2 \log \left (e+f x^2\right )+6 a b f m n x^2 \log \left (e+f x^2\right )+3 b^2 f m n^2 x^2 \log \left (e+f x^2\right )-12 a b f m n x^2 \log (x) \log \left (e+f x^2\right )-6 b^2 f m n^2 x^2 \log (x) \log \left (e+f x^2\right )+6 b^2 f m n^2 x^2 \log ^2(x) \log \left (e+f x^2\right )+12 a b f m x^2 \log \left (c x^n\right ) \log \left (e+f x^2\right )+6 b^2 f m n x^2 \log \left (c x^n\right ) \log \left (e+f x^2\right )-12 b^2 f m n x^2 \log (x) \log \left (c x^n\right ) \log \left (e+f x^2\right )+6 b^2 f m x^2 \log ^2\left (c x^n\right ) \log \left (e+f x^2\right )+6 a^2 e \log \left (d \left (e+f x^2\right )^m\right )+6 a b e n \log \left (d \left (e+f x^2\right )^m\right )+3 b^2 e n^2 \log \left (d \left (e+f x^2\right )^m\right )+12 a b e \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+6 b^2 e n \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+6 b^2 e \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+6 b f m n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 b f m n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^2 f m n^2 x^2 \operatorname {PolyLog}\left (3,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^2 f m n^2 x^2 \operatorname {PolyLog}\left (3,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{12 e x^2} \]
-1/12*(-12*a^2*f*m*x^2*Log[x] - 12*a*b*f*m*n*x^2*Log[x] - 6*b^2*f*m*n^2*x^ 2*Log[x] + 12*a*b*f*m*n*x^2*Log[x]^2 + 6*b^2*f*m*n^2*x^2*Log[x]^2 - 4*b^2* f*m*n^2*x^2*Log[x]^3 - 24*a*b*f*m*x^2*Log[x]*Log[c*x^n] - 12*b^2*f*m*n*x^2 *Log[x]*Log[c*x^n] + 12*b^2*f*m*n*x^2*Log[x]^2*Log[c*x^n] - 12*b^2*f*m*x^2 *Log[x]*Log[c*x^n]^2 + 12*a*b*f*m*n*x^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[ e]] + 6*b^2*f*m*n^2*x^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 6*b^2*f*m* n^2*x^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 12*b^2*f*m*n*x^2*Log[x]* Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 12*a*b*f*m*n*x^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 6*b^2*f*m*n^2*x^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/ Sqrt[e]] - 6*b^2*f*m*n^2*x^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 12* b^2*f*m*n*x^2*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 6*a^2*f*m *x^2*Log[e + f*x^2] + 6*a*b*f*m*n*x^2*Log[e + f*x^2] + 3*b^2*f*m*n^2*x^2*L og[e + f*x^2] - 12*a*b*f*m*n*x^2*Log[x]*Log[e + f*x^2] - 6*b^2*f*m*n^2*x^2 *Log[x]*Log[e + f*x^2] + 6*b^2*f*m*n^2*x^2*Log[x]^2*Log[e + f*x^2] + 12*a* b*f*m*x^2*Log[c*x^n]*Log[e + f*x^2] + 6*b^2*f*m*n*x^2*Log[c*x^n]*Log[e + f *x^2] - 12*b^2*f*m*n*x^2*Log[x]*Log[c*x^n]*Log[e + f*x^2] + 6*b^2*f*m*x^2* Log[c*x^n]^2*Log[e + f*x^2] + 6*a^2*e*Log[d*(e + f*x^2)^m] + 6*a*b*e*n*Log [d*(e + f*x^2)^m] + 3*b^2*e*n^2*Log[d*(e + f*x^2)^m] + 12*a*b*e*Log[c*x^n] *Log[d*(e + f*x^2)^m] + 6*b^2*e*n*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 6*b^2* e*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] + 6*b*f*m*n*x^2*(2*a + b*n + 2*b*Lo...
Time = 0.57 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2825, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -2 f m \int \left (-\frac {b^2 n^2}{4 x \left (f x^2+e\right )}-\frac {b \left (a+b \log \left (c x^n\right )\right ) n}{2 x \left (f x^2+e\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x \left (f x^2+e\right )}\right )dx-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 f m \left (-\frac {b n \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac {b n \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac {\log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{8 e}-\frac {b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e}{f x^2}\right )}{8 e}+\frac {b^2 n^2 \log \left (e+f x^2\right )}{8 e}-\frac {b^2 n^2 \log (x)}{4 e}\right )-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}\) |
-1/4*(b^2*n^2*Log[d*(e + f*x^2)^m])/x^2 - (b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(2*x^2) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(2*x^2 ) - 2*f*m*(-1/4*(b^2*n^2*Log[x])/e + (b*n*Log[1 + e/(f*x^2)]*(a + b*Log[c* x^n]))/(4*e) + (Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n])^2)/(4*e) + (b^2*n^2* Log[e + f*x^2])/(8*e) - (b^2*n^2*PolyLog[2, -(e/(f*x^2))])/(8*e) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x^2))])/(4*e) - (b^2*n^2*PolyLog[3, -(e /(f*x^2))])/(8*e))
3.2.2.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* (a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r] u, x] - Simp[f*m*r Int[x^(m - 1)/(e + f*x^m) u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m , n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 75.81 (sec) , antiderivative size = 5175, normalized size of antiderivative = 18.75
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}} \,d x } \]
-1/4*(2*b^2*log(x^n)^2 + (n^2 + 2*n*log(c) + 2*log(c)^2)*b^2 + 2*a*b*(n + 2*log(c)) + 2*a^2 + 2*(b^2*(n + 2*log(c)) + 2*a*b)*log(x^n))*log((f*x^2 + e)^m)/x^2 + integrate(1/2*(2*b^2*e*log(c)^2*log(d) + 4*a*b*e*log(c)*log(d) + 2*a^2*e*log(d) + (2*(f*m + f*log(d))*a^2 + 2*(f*m*n + 2*(f*m + f*log(d) )*log(c))*a*b + (f*m*n^2 + 2*f*m*n*log(c) + 2*(f*m + f*log(d))*log(c)^2)*b ^2)*x^2 + 2*((f*m + f*log(d))*b^2*x^2 + b^2*e*log(d))*log(x^n)^2 + 2*(2*b^ 2*e*log(c)*log(d) + 2*a*b*e*log(d) + (2*(f*m + f*log(d))*a*b + (f*m*n + 2* (f*m + f*log(d))*log(c))*b^2)*x^2)*log(x^n))/(f*x^5 + e*x^3), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \]